Systems of polynomial equations defining hyperelliptic d-osculating covers

Let X denote a fixed smooth projective curve of genus 1 defined over an algebraically closed field of arbitrary characteristic p not equal 2. For any positive integer n, we consider the moduli space H(X, n) of degree-n finite separable covers of X by a hyperelliptic curve with three marked Weierstrass points. We parameterize H(X, n) by a suitable space of rational fractions and apply it to studying the (finite) subset of degree-n hyperelliptic tangential covers of X. We find a polynomial characterization for the corresponding rational fractions and deduce a square system of polynomial equations whose solutions parameterize these covers. Furthermore, we also obtain nonsquare systems parameterizing hyperelliptic d-osculating covers for any d > 1.